It depends on how you plan to tour.
If you're doing fully-loaded touring using panniers with at least an average amount of gear, it will be difficult to find a suitable bike where you are going. And it might cost enough that you won't want to simply leave it behind when you go home. So, despite the hassle, you might want to take your own.
If you're doing light-weight touring with minimal gear, then you might be able to find a $100 WalMart bike that will do.
If you plan to use a trailer, you could take the trailer with you and attach it to a rental bike.
You didn't say how long your trip is. If it's only a week, then you can probably get by without a perfect bike fit. But if it's longer than a week, then using a bike you have a lot of experience with and know is comfortable for you is worth a lot.
The questions you're asking have bugged many of us before, and after wrestling with the options, most of us end up taking our own bikes with us.
If you are bringing your bike, I find arriving at your departure point about 90 minutes before your tour begins is plenty. If you are not bringing your bike, it would probably be wise to arrange the purchase or rental before you go so it will be all ready for you when you arrive. I'm not sure why you would want to arrive days early unless you want to tour around the departure city.
You didn't ask for this advice, but here goes. I think it would be wise to do your first trip starting and ending from home. That eliminates all the rest of your questions.